Finite-Element Modeling Step By Step

Software for performing meaningful analysis increasingly shows up on the desktops of design engineers. A finite-element model can be thought of as a system of solid springs. When a load is applied to the structure, all elements deform until all forces balance. For each element in the model, equations can be written relating displacements and forces at the nodes. The element shown here, for example, is a 2D quadrilateral having four nodes. Each node has two degrees of freedom associated with it (displacements in X and Y directions), so that the element has a total of eight degrees of freedom. There must also be a nodal force for each nodal degree, so there are also eight nodal forces for the element. These displacements and forces are identified by a coordinate numbering system for entry the computer program. For example, dxi1 is the deflection in the X direction for element i at node 1, while dyi1 is the deflection in the Y direction for the same node in the same element. Forces are identified in a similar manner, so that Fxi1 is the force in the X direction for element i at node 1. An equation relating displacements and forces for the element takes the form of basic spring equation, F = kd. For four nodes:

• k₁₁d_xi1 + k₁₂d_yi1 + k₁₃d_xi2 + k₁₄d_yi2 + . . . + k₁₈d_yi4 = F_xi1
   
• k₂₁d_xi1 + k₂₂d_yi1 + k₂₃d_xi2 + k₂₄d_yi2 + . . . + k₂₈d_yi4 = F_yi1
   
• k₃₁d_xi1 + k₃₂d_yi1 + k₃₃d_xi2 + k₃₄d_yi2 + . . . + k₃₈d_yi4 = F_xi2
   
• k₈₁d_xi1 + k₈₂d_yi1 + k₈₃d_xi2 + k₈₄d_yi2 + . . . + k₈₈d_yi4 = F_yi4
   

The k factors are stiffness coefficients relating the nodal deflections and forces, and are calculated by the finite-element program from material properties such as Young's modulus and Poisson's ratio, and from the element geometry. Thus, in the example, coefficient k₁₃ relates deflection 3 and force 1. If degrees of freedom and nodal forces are consecutively numbered (d_xi1 = d₁, d_yi1 = d₂, F_xi1 = F₁, F_yi1 = F₂, and so forth), the matrix can be renumbered to show how stiffness coefficients relate nodal forces and deflections. When a structure is modeled, individual sets of matrix equations are automatically generated for each element. The elements in the model share common nodes so individual sets of matrix equations can be combined into a global set of matrix equations. This global set relates all the nodal degrees of freedom to the nodal forces, and the nodal degrees of freedom are solved simultaneously from the global matrix. When displacements for all nodes are known, the state of deformation of each element is known. And, when deformation of each element is know, the stress and strain within the element are also known. For simple static analysis, the finite-element method is a two-step process. Nodal displacements are first simultaneously calculated from the element stiffness and the nodal forces, both internal and external. Next, stresses are calculated, generally at the each element's centroid. Because displacements are calculated for only a finite number of points in the structure, the finite-element method is a numerical approximation rather than an exact solution.